# A line segment goes from #(1 ,2 )# to #(4 ,1 )#. The line segment is reflected across #x=-1#, reflected across #y=3#, and then dilated about #(2 ,2 )# by a factor of #3#. How far are the new endpoints from the origin?

Given points

Reflected across

New coordinates after reflection are

Now we we have to find A”, B” after rotation about point C (2,2) with a dilation factor of 3.

Similarly,

New endpoints are

Distance of new points from origin

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The new endpoints of the line segment after being reflected across x=-1, reflected across y=3, and then dilated about (2, 2) by a factor of 3 are:

Endpoint 1: Coordinates after reflection across x=-1: (-3, 2) Coordinates after reflection across y=3: (-3, 6) Coordinates after dilation about (2, 2) by a factor of 3: (-3, 12)

Endpoint 2: Coordinates after reflection across x=-1: (6, 1) Coordinates after reflection across y=3: (6, 5) Coordinates after dilation about (2, 2) by a factor of 3: (12, 9)

To find the distance of each endpoint from the origin, we use the distance formula: √((x - x₁)² + (y - y₁)²)

For Endpoint 1: Distance = √((-3 - 0)² + (12 - 0)²) = √(9 + 144) = √153 ≈ 12.37 units

For Endpoint 2: Distance = √((12 - 0)² + (9 - 0)²) = √(144 + 81) = √225 = 15 units

Therefore, the distance of the new endpoints from the origin are approximately 12.37 units and 15 units, respectively.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- A line segment goes from #(1 ,1 )# to #(4 ,2 )#. The line segment is reflected across #x=2#, reflected across #y=-1#, and then dilated about #(1 ,1 )# by a factor of #2#. How far are the new endpoints from the origin?
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